3. The attempt at a solution
I don't really understand how to prove that there is a linear transformation with these coordinates. I think I begin by defining another arbitrary, linearly independent, transformation such as T(1,0,0)=(1,1). Then I don't really know where to go from here.

I think I begin by defining another arbitrary, linearly independent, transformation such as T(1,0,0)=(1,1).

Think about how this helps you along with the requirement that T is a linear transformation. For example, what does your requirement that having the transformation of another linearly independent vector in R3 give you?

Think about how this helps you along with the requirement that T is a linear transformation. For example, what does your requirement that having the transformation of another linearly independent vector in R3 give you?

Well, you need to define what your ##\alpha## and ##\beta## etc are. You have three vectors with which you can express any vector in R3 as a linear combination. What will be the resulting transformation of such a linear combination? Will it fulfil the requirements you have?

No, not if you already have fixed it for three vectors. Use the linear property of T! Let us say you have the vectors ##v_1, v_2, v_3## and have fixed ##T(v_i) = u_i## for ##i = 1,2,3##. Now you take a linear combination of those ##v = \sum_i c_i v_i##. What is ##T(v)##?

Yes!!! it does! Thank you so much. It makes much more sense now. I just have one other question, why do the vectors need to span all of R^3 in the first place? Is it so the transformation can go to any place in R^2?

Unless you specify the transformation for any arbitrary vector in R3 you have not really found a linear transformation from R3 to R2 but only from a two-dimensional subspace. A more complete answer would be: Yes, it exists, but is not unique. (You could have selected any vector in R2 as the image of your third linearly independent vector in R3).